Using your Head is Permitted

July 2007 riddle

This month's riddle follows an original suggestion by Moshe Wolf.
Prove either or both parts for your name to be mentioned as a solver.
A separate list of solvers will be kept for each of the two parts.

Part 1:

Let n be a positive integer such that 2n-1 is a digit anagram of
n (in its decimal representation).
In other words, n and 2n-1 share the same digits, but in a
different order.
An example of such a pair would be n=37, 2n-1=73.

Show that if n's least significant digit is 3, n must also
contain an 8.

Part 2:

Prove that for every positive integer m there exists a positive integer
k, such that 2km is a digit anagram of km.

List of solvers:

Part 1:

Itsik Horovitz (2 July 18:00)

Part 2:

Itsik Horovitz (2 July 18:00) Or Sheffet (24 July 22:00)

Elegant solutions can be submitted to the puzzlemaster at riddlesbrand.scso.com.
Names of solvers will be posted on this page. Notify if you don't want
your name to be mentioned.